Subgroups — C₁₃×ΓL(1,𝔽₈)×Dic₁₉ #35208806

Order 20748.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₇×C₁₃×Dic₁₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₁₃×C₁₉×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
4	C₂×C₇×C₁₃×C₁₉	C₂×C₃	1	1	yes	characteristic
5	C₃×C₁₃×Dic₁₉	—	7	1	no	maximal, Hall
6	C₇×C₁₃×C₁₉	C₃×C₄	1	1	yes	characteristic, complemented, Hall
7	ΓL(1,𝔽₈)×Dic₁₉	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₁₃×C₁₉	—	7	1	no	—
9	C₄×C₁₃×ΓL(1,𝔽₈)	—	19	1	no	maximal, Hall
10	C₁₃×Dic₁₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₁₉×ΓL(1,𝔽₈)	C₂×C₁₃	1	1	yes	characteristic
12	C₃×C₁₃×C₁₉	—	7	1	no	Hall
13	C₂×C₁₃×ΓL(1,𝔽₈)	D₁₉	1	1	yes	characteristic
14	C₇×Dic₁₉	C₃×C₁₃	1	1	yes	characteristic, complemented, Hall
15	C₂×C₁₃×C₁₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
16	C₁₉×ΓL(1,𝔽₈)	C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
17	C₄×C₇×C₁₃	—	19	1	no	Hall
18	C₁₃×ΓL(1,𝔽₈)	Dic₁₉	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₇×C₁₉	C₂×C₃×C₁₃	1	1	yes	characteristic
20	C₁₃×C₁₉	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
21	C₃×Dic₁₉	—	7	1	no	Hall
22	C₂×C₇×C₁₃	C₃×D₁₉	1	1	yes	characteristic
23	C₃×C₄×C₁₃	—	133	1	no	Hall
24	C₇×C₁₉	C₃×C₄×C₁₃	1	1	yes	characteristic, complemented, Hall
25	C₂×C₃×C₁₉	—	7	1	no	—
26	C₇×C₁₃	C₃×Dic₁₉	1	1	yes	characteristic, complemented, Hall
27	C₄×ΓL(1,𝔽₈)	—	19	1	no	Hall
28	C₂×C₃×C₁₃	—	7	1	no	—
29	Dic₁₉	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
30	C₃×C₁₉	—	7	1	no	Hall
31	C₄×C₁₃	—	19	1	no	Hall
32	C₂×ΓL(1,𝔽₈)	C₁₃×D₁₉	1	1	yes	characteristic
33	C₃×C₁₃	—	7	1	no	Hall
34	C₂×C₁₉	C₂×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
35	C₄×C₇	—	19	1	no	Hall
36	C₂×C₁₃	ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic
37	ΓL(1,𝔽₈)	C₁₃×Dic₁₉	1	1	yes	characteristic, direct product factor, Hall
38	C₁₉	C₄×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₇	C₃×C₁₃×D₁₉	1	1	yes	characteristic
40	C₁₃	ΓL(1,𝔽₈)×Dic₁₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₃×C₄	—	133	1	no	Hall
42	C₇	C₃×C₁₃×Dic₁₉	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	7	1	no	—
44	C₄	—	19	1	no	Sylow
45	C₃	—	7	1	no	Sylow
46	C₂	C₁₃×ΓL(1,𝔽₈)×D₁₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₃	ΓL(1,𝔽₈)×D₁₉
commutator, hypocenter	C₇×C₁₉	C₃×C₄×C₁₃
Frattini	C₂	C₁₃×ΓL(1,𝔽₈)×D₁₉
Fitting, socle	C₂×C₇×C₁₃×C₁₉	C₂×C₃
prime radical	C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	C₂
cosocle	C₂×C₇×C₁₉	C₂×C₃×C₁₃
LCS	G ▷ C₇×C₁₉
UCS	C₁ ◁ C₂×C₁₃

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×C₁₃×Dic₁₉	C₃	semidirect
3. C₁₃×C₁₉×ΓL(1,𝔽₈)	C₄	semidirect
6. C₇×C₁₃×C₁₉	C₃×C₄	semidirect
7. ΓL(1,𝔽₈)×Dic₁₉	C₁₃	direct (N × H)
10. C₁₃×Dic₁₉	ΓL(1,𝔽₈)	direct (N × H)
14. C₇×Dic₁₉	C₃×C₁₃	semidirect
16. C₁₉×ΓL(1,𝔽₈)	C₄×C₁₃	semidirect
18. C₁₃×ΓL(1,𝔽₈)	Dic₁₉	direct (N × H)
20. C₁₃×C₁₉	C₄×ΓL(1,𝔽₈)	semidirect
24. C₇×C₁₉	C₃×C₄×C₁₃	semidirect
26. C₇×C₁₃	C₃×Dic₁₉	semidirect
29. Dic₁₉	C₁₃×ΓL(1,𝔽₈)	direct (N × H)
37. ΓL(1,𝔽₈)	C₁₃×Dic₁₉	direct (N × H)
38. C₁₉	C₄×C₁₃×ΓL(1,𝔽₈)	semidirect
40. C₁₃	ΓL(1,𝔽₈)×Dic₁₉	direct (N × H)
42. C₇	C₃×C₁₃×Dic₁₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×C₁₉×ΓL(1,𝔽₈)	2
C₇×C₁₃×Dic₁₉	3
C₃×C₁₃×Dic₁₉	7
ΓL(1,𝔽₈)×Dic₁₉	13
C₄×C₁₃×ΓL(1,𝔽₈)	19

Subgroups — C13×ΓL(1,𝔽8)×Dic19 #35208806

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×ΓL(1,𝔽₈)×Dic₁₉ #35208806